The entrance to the lair is in the middle of a village, at the base of a wall on the inside of a walled-off courtyard. Climb over the buildings to get to it.

Start by running west through a doorway. The hall will eventually bring you to the upper level of the building, from which you can climb over a broken armoire to jump across the hall you started in...but on top of chandeliers. Be careful not to fall from up high. To get full sync in this mission, you can't lose more than one block of health. There are no enemies to fight, but falling from significant distance will basically end your full sync run.

When you reach an area with a statue of a woman with her hands out, climb up onto the statue and jump left onto a hanging lantern. As Ezio is flung off the lantern, you need to manually tap the grab button to latch onto some wood surrounding a column. From there you can shimmy around and continue.

You'll swing down to a lower level before climbing back to the upper level and heading west. In a large room with a floor that gives out as you enter, hug the left (south) wall and use a thin, rickety wooden structure attached to the wall to grab a ledge ahead.

Soon you'll enter the final area, a massive cathedral hall. Run to the north end and interact with a glowing gate. You need to equip a projectile weapon of some sort (bullets, crossbow, throwing knives) to hit a hanging sand bag left of the gate. If you have no ammo, search the area for containers on desks that will replenish your supply.

Shooting the first sand bag opens up a passage to the upper level of the hall. You'll need to make your way to the northwest corner of the upper level before crossing back to the southeast corner. There's a second sand bag to shoot down, causing another reaction.

Use a leap of faith into a straw pile below and use the same passage you opened previously to get back to the second level of the hall. There's a third sandbag in the southwest corner to shoot down. This finally opens up the gate to the north, letting you grab the key from the main treasure chest and climb a ladder down to the lair's exit.